So where is the air intake(s) ? Is it a air breath engine?
I added a photo, didn't you see the air intake? It can be seen under the missile.
The thickest thing seen in the back of the missile in the photo is the booster. Kale's KTJ 1750 engine, which runs on liquid fuel, is positioned before the booster. This engine broke the world record for thrust, by compare weight&size, of course. I am also attaching a test video of the engine.
You can also see the air intake&booster of the missile at 5:00 in this video.
@BON PLAN;
This missile is a potentially very dangerous missile. Because it can be same time launched from many many platforms and it can organize swarm attacks through inter-missiles EW resistant communication.
Yes, maybe one of them alone cannot sink a ship, but 3-4 Çakır missiles hitting the same point will easily sink any ship. This Çakır missile it's a weapon that I certainly wouldn't want my enemies to have! To call it "a toy/firecracker" would be a big fatal mistake.
By the way, the warhead carried by the missile is 70 kg around. Even if a ship that is hit by only one Çakır missile does not sink, she will be out of war.